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QUESTION 4. The Magellanic Stream orbits the Milky Way and extends from about 50 kpc to 100 kpc from the Galactic center. (4a) A clump of gas in the Magellanic Stream moves in a circular orbit with orbital speed 245 km/s and radius 75 kpc about the Galactic center. Treating the clump of gas as a point mass, estimate the mass of the Milky Way within 75 kpc. If the total B-band luminosity of the Milky Way is LB tot = 2.3 x 10°Lo, what is the implied mass-to-light ratio? (12 marks] (4b) A clump of gas at the far tip of the stream initially has zero radial velocity when it is 100 kpc from the center of the Milky Way, and has reached an inward radial velocity of 220 km/s after it has fallen to 50 kpc from the Galactic center, without any significant change in its orbital velocity. Estimate the mass of the Milky Way within 50 kpc and the corresponding mass-to-light ratio. [12 marks] (4c) How do the mass-to-light ratios you derived compare with the approximate mass-to-light ratio for standard "luminous” matter? Comment on what your results indicate about the distribution of dark matter in the Milky Way. [6 marks)